Can games be solved without an evaluation function?
Human chess and go experts clearly use evaluation functions. They do come up with moves that look sensible without evaluating the board position, but to validate these candidate moves they evaluate board positions that occur at the end of the variations they calculate. Pretty similar to AlphaGo. 

Inputting two board states and outputting a preference is a (much) more complex task than mapping one board state into the real numbers. And it gives you less information. So its a lose-lose choice. (I did try something very similar and it didn't work at all. The reason is that you didn't just double the input size, rather you made the input space quadratically bigger.)

If you compare two board states that differ just by one move, then your input space doesn't quite explode as much, but you have to do a ton of comparisons to make a decision. The logical choice would be to output a preference distribution over all possible moves - but that's exactly what AlphaGo does in its policy network. There is also an earlier paper which trained a network to predict expert moves, which comes down to the same thing. And yes, both these networks play quite strongly without any search or board evaluation. But nowhere near AlphaGo-level.